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Quantum Information Processing

Erschienen in:

01.10.2021

Weak measurements, non-classicality and negative probability

verfasst von: Sooryansh Asthana, V. Ravishankar

Erschienen in: Quantum Information Processing | Ausgabe 10/2021

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Abstract

This paper establishes a direct, robust, and intimate connection among (i) non-classicality tests for various quantum features, e.g. non-Boolean logic, quantum coherence, nonlocality, quantum entanglement, quantum discord; (ii) negative probability, and, (iii) anomalous weak values. It has been shown (Adhikary et al. Eur Phys J D 74(68):68, 2020; Asthana et al. Quantum Inform Process 20(1):1–33, 2021) that the nonexistence of a classical joint probability scheme gives rise to sufficiency conditions for nonlocality, a nonclassical feature not restricted to quantum mechanics. The conditions for nonclassical features of quantum mechanics are obtained by employing pseudo-probabilities, which are expectation values of the parent pseudo-projections. The crux of the paper is that the pseudo-probabilities, which can take negative values, can be directly measured as anomalous weak values. We expect that this opens up new avenues for testing nonclassicality via weak measurements and also gives deeper insight into negative pseudo-probabilities, which become measurable. A quantum game, based on violation of a classical probability rule, is also proposed that can be played by employing weak measurements.

Vorheriger Artikel Reversible optical–microwave quantum conversion assisted by optomechanical dynamically dark modes

Nächster Artikel Partial orbits of quantum gates and full three-particle entanglement

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Entanglement inequalities are violated by all the separable states and obeyed by at least one entangled state.

For example, in quantum mechanics, probability amplitudes add, and not the probabilities.

For the sake of brevity, we represent both observables and operators by the same symbol throughout.

For example, the most general PP representing the joint outcome, $O_1=o_1, O_2=o_2$ and $O_3=o_3$ is,

$$\begin{aligned} {\varvec{\Pi }}=\nu _1{\varvec{\Pi }}_1+\nu _2{\varvec{\Pi }}_2+\nu _3{\varvec{\Pi }}_3;~0\le \nu _i\le 1;~\sum _{i=1}^3\nu _i=1, \end{aligned}$$

where,

$$\begin{aligned} {\varvec{\Pi }}_{1}=\dfrac{1}{2}(\pi _{o_1}\pi _{o_2}\pi _{o_3}+\mathrm{h.c.});~{\varvec{\Pi }}_{2}=\dfrac{1}{2}(\pi _{o_2}\pi _{o_1}\pi _{o_3}+\mathrm{h.c.});~{\varvec{\Pi }}_{3}=\dfrac{1}{2}(\pi _{o_1}\pi _{o_3}\pi _{o_2}+\mathrm{h.c.}). \end{aligned}$$

This situation can occur elsewhere as well. For example, nonlocality can be detected by both—inequalities and Hardy-type paradoxes [55].

The unit pseudoprojection ${\varvec{\Pi }}_{a_1a_2a_1a_3}$ is given as,

$$\begin{aligned} {\varvec{\Pi }}_{a_1a_2a_1a_3} =\frac{1}{2}(\pi _{a_2}\pi _{a_1}\pi _{a_1}\pi _{a_3}+\pi _{a_3}\pi _{a_1}\pi _{a_1}\pi _{a_2})= \frac{1}{2}(\pi _{a_2}\pi _{a_1}\pi _{a_3}+\pi _{a_3}\pi _{a_1}\pi _{a_2}). \end{aligned}$$

Even the completely mixed two-dimensional state has a negative pseudo-probability for the joint event, when ${\varvec{\sigma }}\cdot \hat{a}_1, {\varvec{\sigma }}\cdot \hat{a}_2, {\varvec{\sigma }}\cdot \hat{a}_3$ take value $+1$, where $\hat{a}_1, \hat{a}_2, \hat{a}_3$ are coplanar and at an included angle of $\frac{2\pi }{3}$. The completely symmetrised PP can be constructed as follows:

$$\begin{aligned} {\varvec{\Pi }}_{a_1a_2a_3}&=\dfrac{1}{3!}(\pi _{a_1}\pi _{a_2}\pi _{a_3}+\pi _{a_1}\pi _{a_3}\pi _{a_2}+\pi _{a_2}\pi _{a_1}\pi _{a_3}+\pi _{a_2}\pi _{a_3}\pi _{a_1}+\pi _{a_3}\pi _{a_1}\pi _{a_2}+\pi _{a_3}\pi _{a_2}\pi _{a_1})\\&=-\dfrac{1}{16}, \end{aligned}$$

whose overlap with the completely mixed state is negative, $-\frac{1}{16}$.

Schrödinger, E.: Quantisation as a problem of proper values (part iv). Annalen der Physik 81, 102–123 (1926)

Heisenberg, W: Zeitschrift für Physik 33, 879–893 (1925)

Dirac, P.A.M.: The physical interpretation of quantum dynamics. Proc. Roy. Soc. Lond. A 113, 621–641 (1927)ADSMATHCrossRef

Feynman, R.P.: Space time approach to non relativistic quantum mechanics. Rev. Mod. Phys 20, 367–387 (1948)ADSMathSciNetMATHCrossRef

Bohm, D.: A suggested interpretation of quantum theory in terms of hidden variables. Phys. Rev. 85, 166–179 (1952)ADSMathSciNetMATHCrossRef

Nelson, E.: Derivation of the schrodinger equation from newtonian mechanics. Phys. Rev. 150, 1079 (1966)ADSCrossRef

Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)ADSMATHCrossRef

Schrödinger, E.: Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften 23(50), 844 (1935)ADSMATHCrossRef

Bell, J.S.: On the einstein podolsky rosen paradox. Phys. Physique Fizika 1, 195–200 (1964)MathSciNetCrossRef

10.

Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17(1), 59–87 (1967)MathSciNetMATH

11.

Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351–1354 (1988)ADSCrossRef

12.

Vaidman, L.: In Greenberger, D., Hentschel, K., Weintert, F. (eds.) Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy. Springer Science & Business Media (2009)

13.

Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)ADSMathSciNetMATHCrossRef

14.

Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)ADSMATHCrossRef

15.

Knill, E., Laflamme, R.: Power of one bit of quantum information. Phys. Rev. Lett. 81, 5672–5675 (1998)ADSCrossRef

16.

Shi, H.-L., Liu, S.-Y., Wang, X.-H., Yang, W.-L., Yang, Z.-Y., Fan, H.: Coherence depletion in the grover quantum search algorithm. Phys. Rev. A 95, 032307 (2017)ADSMathSciNetCrossRef

17.

Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on einstein-podolsky-rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992)ADSMathSciNetMATHCrossRef

18.

Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)ADSMathSciNetMATHCrossRef

19.

Ekert, A.K.: Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67, 661–663 (1991)

20.

Streltsov, A., Kampermann, H., Bruß, D.: Quantum cost for sending entanglement. Phys. Rev. Lett. 108, 250501 (2012)ADSCrossRef

21.

Somm K., Aiman K., Jebarathinam, C., Dipankar, H.: Remote state preparation using correlations beyond discord. Phys. Rev. A 98, 062320 (2018)

22.

Birkhoff, G., Von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37(4), 823–843 (1936)MathSciNetMATHCrossRef

23.

Fine, A.: Hidden variables, joint probability, and the bell inequalities. Phys. Rev. Lett. 48, 291 (1982)ADSMathSciNetCrossRef

24.

Adhikary, S., Asthana, S., Ravishankar, V.: A unified framework for non-classicality: emergence of non-locality and entanglement. Eur. Phys. J. 74(68), 68 (2020)ADS

25.

Asthana, S., Adhikary, S., Ravishankar, V.: Non-locality and entanglement in multi-qubit systems from a unified framework. Quantum Inform. Process. 20(1), 1–33 (2021)MathSciNetCrossRef

26.

Margenau, H., Hill, R.N.: Correlation between measurements in quantum theory. Prog. Theor. Phys. 26(5), 722–738 (11) (1961)

27.

Barut, A.O., Božić, M., Marić, Z.: Joint probabilities of noncommuting operators and incompleteness of quantum mechanics. Found. Phys. 18(10), 999–1012 (1988)ADSMathSciNetCrossRef

28.

Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)ADSMathSciNetMATHCrossRef

29.

Michał, H., Paweł, H., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223(1), 1–8 (1996)MathSciNetMATH

30.

Gühne, O., Hyllus, P., Bruss, D., Ekert, A., Lewenstein, M., Macchiavello, C., Sanpera, A.: Experimental detection of entanglement via witness operators and local measurements. J. Modern Opt. 50(6–7), 1079–1102 (2003)ADSMATHCrossRef

31.

Vaidman, L.: Protective measurements. 0801, 2761 (2008)

32.

Pusey, M.F.: Anomalous weak values are proofs of contextuality. Phys. Rev. Lett. 113, 200401 (2014)ADSCrossRef

33.

Hosoya, A., Shikano, Y.: Strange weak values. J. Phys. A Math. Theor. 43(38), 385307 (2010)ADSMathSciNetMATHCrossRef

34.

Lundeen, J.S., Bamber, C.: Procedure for direct measurement of general quantum states using weak measurement. Phys. Rev. Lett. 108, 070402 (2012)ADSCrossRef

35.

Bamber, C., Lundeen, J.S.: Observing dirac’s classical phase space analog to the quantum state. Phys. Rev. Lett. 112, 070405 (2014)

36.

Higgins, B.L., Palsson, M.S., Xiang, G.Y., Wiseman, H.M., Pryde, G.J.: Using weak values to experimentally determine “negative probabilities” in a two-photon state with bell correlations. Phys. Rev. A 91, 012113 (2015)

37.

Piacentini, F., Avella, A., Levi, M.P., Gramegna, M., Brida, G., Degiovanni, I.P., Cohen, E., Lussana, R., Villa, F., Tosi, A., Zappa, F., Genovese, M.: Measuring incompatible observables by exploiting sequential weak values. Phys. Rev. Lett. 117, 170402 (2016)ADSCrossRef

38.

Dziewior, J., Knips, L., Farfurnik, D., Senkalla, K., Benshalom, N., Efroni, J., Meinecke, J., Bar-Ad, S., Weinfurter, H., Vaidman, L. (eds.) Universality of local weak interactions and its application for interferometric alignment. Proc. Nat. Acad. Sci. 116(8), 2881–2890 (2019)

39.

Dirac, P.A.M.: Bakerian lecture. the physical interpretation of quantum mechanics. Proc. Roy. Soc. of Lond. A Math. Phys. Eng. Sci. 180(980), 1–40 (1942)

40.

Feynman, R.P.: In Hiley, B., Peat, F. (eds.) Chapter-13, Quantum implications: essays in honour of David Bohm. Taylor & Francis (2012)

41.

Chandler, C., Cohen, L., Lee, C., Scully, M., Wodkiewicz, K.: Quasi-probability distribution for spin-1/2 particles. Found. Phys. 22(7), 867–878 (1992)ADSMathSciNetCrossRef

42.

Adhikary, S., Ravishankar, V.: Quantum logic and non-classicality: an explicit formulation. arXiv arXiv:1710.04371v1 (2017)

43.

Weyl, H.: Quantenmechanik und gruppentheorie. Zeitschrift für Physik 46(1), 1–46 (1927)ADSMATHCrossRef

44.

Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys. 24(3), 379–385 (1994)ADSMathSciNetCrossRef

45.

Wiseman, H.M.: Weak values, quantum trajectories, and the cavity-qed experiment on wave-particle correlation. Phys. Rev. A 65, 032111 (2002)ADSMathSciNetCrossRef

46.

Vaidman, L., Ben-Israel, A., Dziewior, J., Knips, L., Weißl, M., Meinecke, J., Schwemmer, C., Ber, R., Weinfurter, H.: Weak value beyond conditional expectation value of the pointer readings. Phys. Rev. A 96, 032114 (2017)ADSCrossRef

47.

Kirkwood, J.G.: Quantum statistics of almost classical assemblies. Phys. Rev. 44, 31–37 (1933)ADSMATHCrossRef

48.

Barut, A.O.: Distribution functions for noncommuting operators. Phys. Rev. 108, 565–569 (1957)ADSMathSciNetMATHCrossRef

49.

Johansen, L.M., Luis, A.: Nonclassicality in weak measurements. Phys. Rev. A 70, 052115 (2004)ADSCrossRef

50.

Johansen, L.M.: Nonclassical properties of coherent states. Physics Letters A 329(3), 184–187 (2004)ADSMathSciNetMATHCrossRef

51.

Pan, A.K.: Interference experiment, anomalous weak value, and leggett-garg test of macrorealism. Phys. Rev. A 102, 032206 (2020)ADSCrossRef

52.

Streltsov, A., Adesso, G., Plenio, M.B.: Colloquium: Quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017)ADSMathSciNetCrossRef

53.

Aharonov, Y., Colombo, F., Popescu, S., Sabadini, I., Struppa, D.C., Tollaksen, J.: Quantum violation of the pigeonhole principle and the nature of quantum correlations. Proc. Nat. Acad. Sci. 113(3), 532–535 (2016)ADSMathSciNetMATHCrossRef

54.

Reznik, G., Bagchi, S., Dressel, J., Vaidman, L.: Footprints of quantum pigeons. Phys. Rev. Res. 2, 023004 (2020)CrossRef

55.

Hardy, L.: Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71, 1665–1668 (1993)ADSMathSciNetMATHCrossRef

56.

Aharonov, Y., Rohrlich, D.: Quantum Paradoxes: Quantum Theory for the Perplexed. John Wiley & Sons (2008)

57.

Gühne, O.: Characterizing entanglement via uncertainty relations. Phys. Rev. Lett. 92, 117903 (2004)ADSCrossRef

Titel: Weak measurements, non-classicality and negative probability
verfasst von: Sooryansh Asthana
V. Ravishankar
Publikationsdatum: 01.10.2021
Verlag: Springer US
Erschienen in: Quantum Information Processing / Ausgabe 10/2021
Print ISSN: 1570-0755
Elektronische ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-021-03289-5

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